(259g) Koopman System Identification with Noise Modeling

The challenge of modeling nonlinear dynamical systems is compounded by the presence of measurement noise, which obscures the true system dynamics and degrades the accuracy of predictive models. When first-principles models are unavailable, data-driven approaches for system identification, i.e., building mathematical models of dynamical systems using measured data, are needed [1]. Recent data-driven advances leveraging Koopman operator theory enable global linearization of nonlinear dynamics by lifting the state space into a space of observables (functions of system states) [2], where the linear system matrices are typically approximated using dynamic mode decomposition (DMD) [3], [4] or extended DMD (eDMD) [5]. However, these methods fail to explicitly account for stochastic disturbances, instead treating noise by absorbing it into deterministic approximations, leading to degraded predictive performance in noisy environments.

In this work, we propose noisy eDMD (neDMD), a data-driven Koopman modeling approach for nonlinear systems that simultaneously identifies linearized system dynamics and characterizes noise distributions when both the governing equations and noise properties (autocorrelations and distributions) are unknown. The method lifts system states and unmeasured noise into a high-dimensional feature space using a dictionary of candidate observables, yielding an approximated linear system, where the process noise, under such a state lifting, is assumed to be approximated as a corresponding nonlinear transformation of a multidimensional white noise signal.

The identification problem is formulated as a regularized least squares optimization and solved via block coordinate descent, which alternately estimates: (1) System matrices through regression [6], and (2) the noise sequence in the lifted space. Once the system matrices are learned from training data, the model’s noise-segregation performance is evaluated on unseen test data using moving horizon estimation (MHE) [7]. This approach reconstructs the white noise history, enabling optimal future state prediction in a manner analogous to prediction error methods in system identification [8].

Beyond addressing nonlinear system identification through a data-driven Koopman modeling algorithm, establishing a bound on the generalized error of predicting future outputs from state history remains critical for practical implementation. We provide these guarantees by deriving a probabilistic upper bound on this generalized prediction error using Bernstein’s inequality from statistical learning theory. Numerical case studies—including a Van der Pol oscillator and a methanol-ethanol distillation column—demonstrate that neDMD achieves order-of-magnitude improvements in prediction accuracy over standard eDMD by explicitly accounting for noise effects in the lifted dynamics.

References:

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[6] M. O. Williams, M. S. Hemati, S. T. Dawson, I. G. Kevrekidis, and C. W. Rowley, “Extending data-driven Koopman analysis to actuated systems,” IFAC-PapersOnLine, vol. 49, no. 18, pp. 704–709, 2016.

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